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ABCD is a parallelogram and the diagonals \(\overline {AC} \) and \(\overline {BD} \) intersect at 'O'. If E, F, G and H are the mid-point of \(\overline {AO}\), \(\overline {DO}\), \(\overline {CO}\) and \(\overline {BO} \) respectively, then what is the ratio of (EF + FG + GH + HE) and (AD + DC + CB + BA)?

ABCD is a parallelogram and the diagonals \(\overline {AC} \) and \(\overline {BD} \) intersect at 'O'. If E, F, G and H are the mid-point of \(\overline {AO}\), \(\overline {DO}\), \(\overline {CO}\) and \(\overline {BO} \) respectively, then what is the ratio of (EF + FG + GH + HE) and (AD + DC + CB + BA)? Correct Answer 1 ∶ 2

Given:

ABCD is a parallelogram

AE = EO

DF = FO

CG = GO

BH = HO

Concept:

The midpoint theorem: The line segment in a triangle joining the midpoint of two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.

Calculation:

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EH/AB = 1/2      ---- (1)

EF/AD = 1/2      ---- (2)

FG/DC = 1/2      ---- (3)

HG/BC = 1/2      ---- (4)

From equation (1), (2), (3) and (4) 

(EF + FG + GH + HE) ∶ (AD + DC + CB + BA) = 1 ∶ 2 

The correct option is 1 i.e. 1 ∶ 2 

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